How Much of the Earth Can You See at Once?

Foreign Michael here, and here I am, the real Michael. This Michael was created by a brilliant young man named Mitchell, who brought it to me at a meet and greet after Brain Candy Live. It is phenomenal, and obviously the most handsome Jack-in-the-Box ever.

Est is a mountain that's 8.848 kilometers tall. Its size is impressive—or is it? Let's cut Earth, the entire planet, right in half, straight through Everest, and then start zooming out. As you can see, Everest's monumentality quickly disappears against Earth's planetary ginormosity. Compared to Earth's diameter, Everest—in fact, all of Earth's ruggedness—barely registers. Now that can seem surprising, since we're often surrounded by diagrams and maps and globes that exaggerate Earth's topography. Now, there's a good reason to do that, but it leads to a misconception about just how smooth Earth is.

Here's a typical example: a cross-section of the United States that I found on Reddit. The vertical axis spans about ten thousand feet, but the horizontal axis represents nearly 14 million. So, stretched to the same scale, so as to mirror reality, the actual smoothness of the Earth becomes apparent. On this one-foot diameter globe, Everest is a bump about two millimeters high. Feels good, but if the Earth was actually this small, Everest would be a bump only a fifth of a millimeter high. It's ten times taller than it should be.

Only 24 people have seen the Earth with their own eyes as a circle small enough to be looked right at—not as the whole world, but as a little thing suspended alone in space. Further away you are from a ball, the more of its surface you can see. Now, we don't always notice this because, in our day-to-day lives, most of the balls we deal with are so small they're almost always many of their own radii away from us, and the available amount of their surface visible is near a maximum. Or they're so big, like the Earth, that we rarely get far enough away fast enough to notice this property.

But the next time you're near a ball, get close to it. You'll see that as you get nearer, more of its surface disappears behind the horizon. But moving back up will make it available again. For most of us stuck our whole lives on Earth's surface, such an experience is impossible, with nothing around to block your view. Five kilometers, about three miles, is about the furthest you can see. Now, haze can limit your view, and atmospheric refraction can slightly extend it. But for the most part, everything you can see happens within an area of just 80 square kilometers. That's not bad, but it's tiny compared with what there is to see.

The higher up you go, of course, the further away you'll be able to see. That's why it's great to be a satellite. Here's the International Space Station. Yeah, look at that nice big portion of the surface in view. Unfortunately, this isn't to scale. If the Earth were the size of an apple, how far away would the International Space Station orbit? Like this far away? Maybe this far away? Or maybe this far away? Actually, it orbits here, 2.7 millimeters above the surface. That's how far the stem of this apple sticks up. That's not very far.

Oh, here's another fun little two-scale fact: if the Earth were the size of an apple, your eyeball would be about the size of the Moon. We often imagine that from the International Space Station, astronauts see the Earth like this, but they're just not that far away from where they actually orbit. International Space Station residents only see about three percent of Earth's surface at any one time, and that three percent is too wide to all fit within a window. When I was in Pittsburgh, their Science Museum had a mock-up of part of the ISS, and as you can see out of the window, Earth is still quite expansive.

Using special lenses, images can be taken from the ISS that look like circular disc Earths. But the lens is distorting things here; it's fitting much more into the picture. In order to truly witness Earth's entire shape with your own eyes, you would need to either smash your face right up against a window or just be floating outside the station. And even then, you would have to move your head to see from edge to edge.

So how high up do you have to go to see the edges of Earth all at once? And even if you did that, how much would you actually see? How much is there to see? Earth is made of stuff—lots of stuff: water, dirt, rocks, and air, all of which are composed of atoms. Tiny things—so teeny that a single drop of water contains not a million atoms, not a billion atoms, or a trillion, or a quadrillion, or a quintillion, but five sextillion atoms. Earth is made of even more stuff—not a septillion atoms, not an octillion, nonillion, decillion, undecillion, duodecillion, tredecillion—not even a quattuordecillion, but 100 quindecillion atoms.

But since we live only on the surface of our planet, we unfortunately can't see most of those atoms. If the Earth was shaped like a disc, or an icosahedron, or say a cube, or a rectangular prism, or two stellated rhombic dodecahedrons, we could see more of the Earth than we normally can. But as things are, we actually see nearly the least of Earth's matter possible because of all solids, a sphere—which the Earth approximately is—has the smallest surface area to volume ratio. The most stuff inside, and the least stuff outside.

So how many of these 10 to the 50 atoms that make up Earth are on the surface for us to see? That's not an easy question. For one thing, technically, atoms on the surfaces of opaque things like rocks and dirt aren't the only parts involved in their appearances. Subsurface scattering can and does happen regardless. Attempting even a rough approximation is illuminating.

I asked Grant from the YouTube channel Three Blue One Brown for some help, and he pointed out that if you calculate the number of circles with atom-sized radii that could pack optimally cover a sphere with the surface area of Earth, you get about 1.5 times 10 to the 34th. That's a lot of atoms! But then he pointed out that the Earth's surface isn't smooth; its roughness provides extra surface area for atoms to occupy. Without a complete description of the shape of Earth's surface—every mountain and valley, every bump on every rock—this is just going to be hopeless, right?

No! Here's the thing: from far away, they make big rugged shapes. In other words, Earth's surface can be described as a fractal. There is a regularity to its roughness. In fact, mathematicians have even assigned a fractal dimension to Earth's surface: 2.3. Now, to see what that means, I highly recommend Grant's video on fractal dimensions; it's fascinating. Using 2.3 and assuming that it applies from the scale of a human hair up to that of a mountain, Grant found that the number of atoms on Earth's surface changes significantly—up from a power of 34 to a power of 37. That's a thousand times more atoms!

So maybe we shouldn't count Earth's roughness out just yet. It's smooth, but not perfectly. To put that number in perspective, the human body contains about 10 to the 27 atoms—that's 10 powers of 10 less than the surface of the Earth. 10 powers of 10 is 10 billion. There are about seven and a half billion humans, so more or less it can be said that there are the same number of atoms in every human body right now as there are on the surface of the Earth. As I've shown before, all human bodies piled into one place would barely even fill the Grand Canyon, but all human atoms spread across the Earth would almost perfectly cover it, just one atom deep.

Fun fact: the mass of the atmosphere is about 2.5 percent less than what you would get by multiplying sea level pressure—14.7 pounds per square inch—by the surface area of the Earth because Earth's terrain displaces about that much air. Earth's surface is pretty cool, obviously! I mean, it's got lichen, and monster trucks, and an island in a lake, on an island in a lake, on an island. But from down here on its surface, we just can't see that much of it. Your view of Earth is obstructed by lots of opaque things: walls, buildings, trees, rocks, terrain. If Earth was flat, you could see further. But sorry, it's a rough world out there.

Or is it? If you could hold the Earth in your hands like this, how bumpy would it actually feel? We already saw that even our planet's biggest bumps barely register relative to Earth's size. But let's go somewhere famously flat, where relative to our size, terrain rarely gets in the way of seeing lots of the planet—the U.S. state of Kansas. I grew up here and took this footage while driving across the state last year. You can probably see why Kansas is often called flatter than a pancake.

However, although it is famously flat, Kansas is not the flattest U.S. state. In a fantastic piece of research, Jerome Dobson and Joshua Campbell defined "looks flat" like this: if from a given point any part of the terrain within the horizon rises more than 0.32 degrees up—about the height of a 30-meter hill—at the horizon, a typical person would say, "Well, hey, that part's not flat." By cleverly applying this rule to topographical data, they were able to give every state a flatness score. West Virginia was the least flat. Kansas was only the seventh flattest. Delaware, Minnesota, Louisiana, North Dakota, and Illinois are all, by this method, flatter than Kansas, as was the number one flattest state—Florida.

Adam Savage and I had the pleasure of visiting Florida with our Brain Candy Live show this year, and as this footage from atop the King Center in Melbourne, Florida shows, it's pretty gosh dang flat. Now, even though Kansas is not the flattest, it is the state most often ranked flattest when the general population is asked. It is truly scientifically flatter than a pancake. It's been demonstrated, but there's more to the story than that.

In 2003, researchers took a 130 millimeter wide pancake procured from IHOP and analyzed its local reliefs. They found the difference between high and low points was on the order of about two millimeters. If a typical pancake like this was the size of Kansas—5 million times larger—two millimeter high peaks would be 10 kilometer high mountains. In comparison, Mount Everest is only about 8.8 kilometers tall, and Earth's deepest scar, the Marianas Trench, is thought to be just under 11 kilometers deep. So not only is Kansas about as smooth as a pancake, but so is every other state in the Union, and so is the entire world.

If you were a giant holding the planet in your hands like this, you and it would be torn apart by the immense tidal forces created by your gravities. If somehow you could avoid that, though, the planet would feel not much rougher than running your hands over a pancake, but a soggy one, right? I mean, most of Earth's surface is covered in water; your hands would get wet. Or would they? Yes, Earth is covered in liquid, but the depth of that liquid, like the mountains above, just doesn't compare to the total size of the planet.

As it turns out, if the Earth was the size of a typical classroom globe—like this one, one foot in diameter—the volume of water contained in, above, and on it would only be about 14 milliliters. That's this much water! It's kind of hard to believe because at this scale, spreading this much water across all of the ocean's surfaces would be pretty much impossible due to surface tension. But this is it—all of Earth's water compared to all of Earth. Ninety percent of the space on our planet life can live in is in here; the other 10 percent is dry land.

So no, you wouldn't get wrinkly fingers playing with an Earth like this. You could stop it dry just with a paper towel. Despite the incredible area oceans cover on our planet, their depth is just nothing compared to the size of our entire planet. Now, you may have heard it said that if the entire planet were shrunk down to the size of a billiard ball, it would be smoother than a billiard ball. After all we've seen so far, that seems believable. But as it turns out, it's not true.

The misconception stems from the interpretation of the World Pool Billiard Association's rules. Now, according to them, a ball must have a diameter of 2.25 inches plus or minus five thousandths of an inch. Now, some writers have taken this to mean that pits and bumps of five thousandths of an inch are allowed. Proportionately on Earth, that would mean a mountain that was 28 kilometers high. So since Earth has none of those, Earth must be smoother than a billiard ball.

Except if bumps that high were actually allowed on a pool ball, a ball covered with 120 grit sandpaper would be within regulation. Clearly, the five thousandths rule is more about roundness—deviation from a sphere—and not texture. In fact, as microscopic photography has shown, imperfections on regulation balls are only one hundred thousandth of an inch, or about half a micrometer deep. And high scale down to the size of a billiard ball, Earth's Mariana Trench would be 49 micrometers deep.

So Earth is smoother than a pancake, but not smoother than a billiard ball; nor, as XKCD wonderfully showed, is Earth smoother than a bowling ball. But hold on—earlier we were using the word flat; now we're using the word smooth. That distinction is important. You see, the Earth isn't flat like a plane; instead, it curves. It's a ball. Pieces of Earth, like Kansas, might be quite smooth, but they curve along with Earth.

If you were to stand in the middle of Kansas, people on the eastern or western edges of the state would appear to be not level with you, but about 8.1 kilometers below you. That's nearly the height of Everest. And if they stood straight up, they'd be tilted nearly two degrees relative to where you thought up was. Now here's an interesting coincidence: generally speaking, one mile from where you are, Earth curves down about eight inches. One kilometer from where you stand, it curves down about eight centimeters.

Now, the rate of drop due to curvature isn't a linear one; you can't just multiply any distance by eight to get the drop due to curvature. Instead, use an online calculator like the one I've linked down in this video's description. You can put in any distance you want. Anyway, the visibility limit caused by Earth's curvature is your horizon. It encircles you like a visual cage, but it's a cage whose radius is determined by how high up your eyes are. Conan O'Brien, at six foot four inches tall, can see up to five kilometers in any direction, but Snooki, at four foot eight, can only see about 4.3.

To find out how far away your horizon is geometrically, just use the online tools I've put down in the description below. Earth's texture can get in the way of your horizon, but can also cause things beyond the horizon to peek into view. Hey, what's that? Dot-com factors all of this in. Now, if Earth was a smooth sphere, the view from atop Ben Nevis, the highest mountain in the British Isles, would end at the horizon 131 kilometers away—about 80 miles. Such an area would look like this.

But factoring in Earth's ups and downs, here's a more precise boundary of what you can see. Loch Treig, Scottish Gaelic for Lake of Death, is only about 10 miles from the peak. That's within an 80-mile radius, but it can't be seen because terrain in the way blocks it. Parts of the Atlantic Ocean and the North Sea—eight times further away—can be seen; they lie at the limit of Earth's curvature just before it bends the surface out of sight. These spikes extending beyond the geometric horizon are caused by things beyond it that are tall enough to peak above Earth's curvature. In the case of Ben Nevis, this includes high elevation parts of Northern Ireland.

Okay, enough about the surface and what it's like close up. Let's go further away and see more. This will be fun! But there will be a trade-off: the further away you are from something, the smaller it will appear to be. Moving away from Earth will make more area available to see, but that area will take up less of your field of view. It can be difficult to illustrate this in a YouTube video because your field of view—the shape and size of what you can see with your head still—just by moving your eyes around—is about 120 degrees up and down and more than 180 degrees horizontal. A screen is just a window of that space, nowhere close to filling it unless you get uncomfortably close.

To help us visualize large apparent sizes, let's replace the spherical Earth with a flat disc that’s always the same distance from the observer. This disc can be given an apparent size equal to Earth's from any altitude, and the disc can contain on it everything that would fit within your horizons from any altitude. Okay, so standing on the surface, looking straight down, Earth will take up nearly a full 180 degrees of your field of view. With your arms extended straight out parallel to Earth, your fingers will point to the edges of the planet.

Your horizon from 400 kilometers up—about where the ISS orbits—three percent of the Earth's surface is within your horizon, but the Earth will only take up about 140 degrees of your vision. Your fingers would point to Earth's edges if you narrowed your arms' angles each by the width of two outstretched fists. One fist is about 10 degrees across at arm's length. Now you can move your eyes from edge to edge horizontally here, but you can't quite take in the full width vertically.

But for more than twice this altitude, 1,000 kilometers away, Earth is only 120 degrees across. That's one less fist width each. This is perfect; that fits within our narrower vertical field of view. So from a thousand kilometers up—about 620 miles—you can just start to see Earth as a complete disk right in front of you at once. However, only seven percent of Earth fits within the horizon from up here.

A birth taken by satellites this far up, like the Suomi NPP, look kind of weird. I mean, North America doesn't actually take up this much of the globe. Earth's 120-degree width has been compressed to fit in an image much narrower. Compare Africa from its height to the famous Blue Marble picture taken from 45,000 kilometers away. The latter looks more realistic, like looking at a globe on your desk.

Geosynchronous satellites are about 35,000 kilometers high. From their altitude, 43.4 percent—nearly a whole half of Earth's surface—is visible, but the Earth only takes up a meager 17 degrees. You could completely cover it with two outstretched palms. That's incredible! But what about from the Moon? Well, from that far away, Earth is only about two degrees across. You could block it out with your outstretched thumb.

However, you can see more of Earth; you can see further around its curvature. From the Moon, 49 percent of Earth's surface is visible—just 49. If you want to see 50 percent—half of Earth's surface at once—you have to go even further away. In fact, you have to go infinitely far away, which you can't. The most of a sphere you can see at once with your own eyes is just half. But in the real world, way before you were actually infinitely far away, the amount of light reaching you from Earth's surface would become so small and infrequent that you wouldn't be able to see anything at all.

Stars like our Sun are much brighter and bigger than the Earth, but only a handful have, even with our best technology, been resolved as anything larger than just a single point. From one thousandth of a light year away, our own Sun would look like every other star in the sky—a single point to the naked eye, only about as wide as R Doradus, the widest star in our sky, from 91 light years away. The point of our Sun would dim to a level undetectable by the naked eye; it would disappear.

Most of the stars in the night sky you can see with your naked eye are further away than that. We can see them, though, because they're brighter and bigger than our own Sun, which means if there's life out there living in systems around the stars we've marveled at and written stories about since humanity began, chances are we are not part of their constellations or folklore. We're a dark patch in the sky to them—an ignorable emptiness framing other stars, the ones they marvel at, while not knowing we're here or that there's anything here.

And as always, thanks for watching! If you don't follow me on Twitter or Instagram, you are missing out on a treasure trove of premium content, so check that out and know this: I love you!

Oh, and this Vsauce shirt is only available to Curiosity Box subscribers. This shirt comes in the latest box! If you sign up now, you will get this shirt, so long as you sign up before it sells out. The Curiosity Box is good for all brains. It comes to your door four times a year, full of science gear and toys that I want you to have. I want you to hold and learn from it. Also, a portion of the proceeds from every box goes to Alzheimer's research. I'm incredibly proud of it!

But what's going on on this shirt? Well, it's modular multiplication around a circle. We have the numbers 1 to 40 around the outside of a circle connected to their product with the number four. So one is connected to four, two is connected to eight, three is connected to twelve, and so on. Even past 40, you can keep imagining the numbers continuing. For instance, one can become 41, two can become 42, and this emerges the Vsauce V.

Many other shapes can be made by using different multipliers or different numbers around the circle. Mathologer has a fantastic video on this topic, which you should check out. I've linked it down in the description. Thank you for being curious, and as always, thanks for watching!